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Let G be a connected, linear, real reductive Lie group with compact centre. Let $$K<G$$ K < G be maximal compact. For a tempered representation $$\pi $$ π of G , we realise the restriction $$\pi |_K$$ π | K as the K -equivariant index of a Dirac operator on a homogeneous space of the form G / H , for a Cartan subgroup $$H<G$$ H < G . (The result in fact applies to every standard representation.) Such a space can be identified with a coadjoint orbit of G , so that we obtain an explicit version of Kirillov’s orbit method for $$\pi |_K$$ π | K . In a companion paper, we use this realisation of $$\pi |_K$$ π | K to give a geometric expression for the multiplicities of the K -types of $$\pi $$ π , in the spirit of the quantisation commutes with reduction principle. This generalises work by Paradan for the discrete series to arbitrary tempered representations.

A geometric realisation of tempered representations restricted to maximal compact subgroups